Problem: $P(x)=3x^4-2x^3+2x^2-1$ What is the remainder when $P(x)$ is divided by $(x+1)$ ?
Answer: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x+1)$, which can be written as $(x-({-1}))$, is $P({-1})$ : $\begin{aligned} P({-1})&=3({-1})^4-2({-1})^3+2({-1})^2-1 \\\\ &=3\cdot 1-2(-1)+2\cdot 1-1 \\\\ &=6 \end{aligned}$ In conclusion, the remainder when $P(x)$ is divided by $(x+1)$ is $6$.